Adequate representation, definition

The expressions eqs. (1.197), (1.199), (1.200), (1.201) are completely general. From them it is clear that the reduced density matrices are much more economical tools for representing the electronic structure than the wave functions. The two-electron density (more demanding quantity of the two) depends only on two pairs of electronic variables (either continuous or discrete) instead of N electronic variables required by the wave function representation. The one-electron density is even simpler since it depends only on one pair of such coordinates. That means that in the density matrix representation only about (2M)4 numbers are necessary to describe the system (in fact - less due to antisymmetry), whereas the description in terms of the wave function requires, as we know n 2m-n) numbers (FCI expansion amplitudes). However, the density matrices are rarely used directly in quantum chemistry procedures. The reason is the serious problem which appears when one is trying to construct the adequate representation for the left hand sides of the above definitions without addressing any wave functions in the right hand sides. This is known as the (V-representability problem, unsolved until now [51] for the two-electron density matrices. The second is that the symmetry conditions for the electronic states are much easier formulated and controlled in terms of the wave functions (Density matrices are the entities of the second power with respect to the wave functions so their symmetries are described by the second tensor powers of those of the wave functions). 

Conventional databases are not sufficient for an adequate representation of the relevant product and process knowledge. On the one hand, current systems do not consider the dynamic of the development process sufficiently. On the other hand, it is not possible to assess the consequences of one s definition. However, this is a fundamental prerequisite for effective cooperation. 

Whilst the above is perfectly adequate for the description of processes observed with continuous-wave (cw) input, proper representation of the optical response to pulsed laser radiation requires one further modification to the theory. It is commonly thought difficult to represent pulses of light using quantum field theory indeed, it is impossible if a number state basis is employed. However by expressing the radiation as a product of coherent states with a definite phase relationship, it is relatively simple to construct a wavepacket to model pulsed laser radiation [39]. The physical basis for this approach is that pulses necessarily have a finite linewidth and therefore in fact entail a large number of radiation modes, so that for the pump radiation, it is appropriate to construct a coherent superposition... 

UVCB substances are substances of unknown or variable composition, complex reaction products, and biological materials that cannot be represented by unique structures and molecular formulas. Some UVCB substances are not adequately described by their CA Names and have supplemental definitions that are considered integral parts of the names for TSCA purposes. The guidance, entitled Toxic Substances Control Act Inventory Representation For Chemical Substances Of Unknown Or Variable Composition, Complex Reaction Products And Biological Materials UVCB Substances points out that any substance that matches a CA Name but is not covered by the substance description is not considered to be covered by that Inventory listing. 

The basic approximations made in arriving at the reactor point effectiveness are (1) isothermal pellet, (2) negligible external mass transfer resistance, and (3) estimation of the pellet center concentration by a simple relationship when the reaction is not severely diffusion-limited. The first two approximations are quite adequate in view of the fact that the mass Biot number is of the order of hundreds under realistic reaction conditions. Both theoretical and experimental justifications for these approximations have been given in Chapter 4. The first approximation will be relaxed when reactions affected by pore-mouth poisoning are considered since a definite temperature gradient then exists within the pellet. An additional approximation is the representation of the difference between the Arrhenius exponentials evaluated at the pellet surface and the bulk-fluid temperatures by a linear rela-... 

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